These follow by observing that the linearization of the entropy term is
\(\int _{{\mathrm {P}}} {\rm tr} \Big [ \big ({\rm Hess}(u) + \sqrt{-1}B\big )^{-1} \circ {\rm Hess}(\dot{u})\Big ] {\mathrmsl d}x = \int _{{\mathrm {P}}} {\rm tr}\big ({\bf X} \circ {\rm Hess}(\dot{u})\big ) {\mathrmsl d}x.\) 
Using that \({\bf F}\)  is linear, and differentiating one more time the above expression yields the formula for the Hessian of \({\bf M}\) . To obtain the first formula, we need to integrate by parts the RHS of the above expression as in [1]}. To this end, we observe that \({\bf X}= \big [{\rm Hess}(u) + B \big ({\rm Hess}(u)\big )^{-1}B\big ]^{-1}\)  is a smooth symmetric matrix on \({\mathrm {P}}\) , which satisfies the first order boundary conditions of  (see also Remark 4 in that reference). The second variation follows easily by passing another \(s\)  derivative through the variation formula above and using that \(\ddot{u} = 0\) .
